The degree distribution of the random multigraphs

Abstract

In this paper, as a generalization of the binomial random graph model, we define the model of multigraphs as follows: let G(n; {p _k }) be the probability space of all the labelled loopless multigraphs with vertex set V = {υ ₁, υ ₂, …, υ _n }, in which the distribution of $t_{v_i ,v_j }$, the number of the edges between any two vertices υ _i and υ _j is

$P\{ t_{v_i ,v_j } = k\} = p_k ,k = 0,1,2,...$

and they are independent of each other. Denote by X _d = X _d (G), Y _d = Y _d (G), Z _d = Z _d (G) and Z _cd = Z _cd (G) the number of vertices of G with degree d, at least d, at most d and between c and d. In this paper, we discuss the distribution of X _d , Y _d , Z _d and Z _cd in the probability space G(n; {p _k }).